Analysis of hyperbolic-type partial differential equations for non-fourier type heat conduction models

Abstract

Heat transfer modelling is routinely used to model the interaction between a heat source and a material specimen in applications such as additive manufacturing and medical surgery.

The Fourier heat conduction model is well-known in the field of heat transfer, but in cases involving ultra-short heat pulses, or extremely small specimens, alternative models such as the Cattaneo-Vernotte \mbox{(C-V)} and dual-phase-lag (DPL) models are proposed. These two models are based on the concept of lagging responses (or lag times) in the heat flux and the temperature gradient.

In 1982 an article appeared that reported on the existence of unwanted oscillations related to a so-called ``benchmark" problem that is based on the \mbox{C-V} model. This problem was studied and it was shown that the unwanted oscillations is the result of an ill-posed problem and not due to the choice of the numerical technique used to solve the problem. The problem was re-formulated to have a smooth initial condition and divided into auxiliary problems. It was solved using D'Alembert's and the finite element method, resulting in an oscillation-free solution.

The theory and terminology of vibration analysis, \emph{e.g. overdamped and underdamped modes}, were incorporated into the Fourier, \mbox{C-V} and DPL heat conduction models. Weak variational formulations of these models, in terms of bilinear forms, were presented and the well-posedness of the model problems was established, based on a general existence result published in 2002.

The modal analysis method was applied to the model problems and formal series solutions were derived. Convergence of the series solutions was proved in terms of the energy and inertia norms. This was used as a guideline to ensure accurate approximations for the series solutions of the model problems.

Realistic lag time values were derived using modal analysis. This relied on the assumption that the solutions for the \mbox{C-V} and Fourier models will be the same after a sufficiently long time.

The concept of a \emph{wane time} was introduced as the time instant at which the Fourier and \mbox{C-V} model predictions will correspond. This was proved with numerical experiments based on a continuous-heating model problem.

Two model problems, based on single- and multi-pulse heating, were used to study aspects such as the contribution of overdamped and underdamped modes to the predicted temperature, the influence of the lag time values on the \mbox{C-V} and DPL model predictions, and the effect of heating parameters, \emph{e.g.} the duty ratio and the number of heating pulses on the model predictions.

In conclusion, modal analysis proved to be successful in determining reliable lag times values and was effective for the numerical investigations into the properties of the solutions of the model problems. Future research should focus on investigating model problems that resemble reliable experimental techniques, thereby facilitating comparison of theory with practice.